Check for a Biased Wheel, or a Dealer’s biased spin with Chi Square Tests

The Chi2 distribution tests for the difference between the observed and the expected in terms of frequencies. We can apply this to a simple example:

In double-zero roulette, we have 38 numbers and the expected probability of any one of these numbers appearing is 1/38 or 0.026316 or 2.6316%. Now, assuming you track all the results that appear on your roulette tables, you’d be able to check for biased wheels or even if your dealers have developed the muscle memory to spin at a regular area of the wheel.

As with all things probable, do note that nothing is impossible. It may be unlikely, but never impossible. Always correlate your findings with footage from surveillance.

Number

Probability

0-0

0.026316

0

0.026316

1

0.026316

2

0.026316

3

0.026316

4

0.026316

5

0.026316

6

0.026316

7

0.026316

8

0.026316

9

0.026316

10

0.026316

11

0.026316

12

0.026316

13

0.026316

14

0.026316

15

0.026316

16

0.026316

17

0.026316

18

0.026316

19

0.026316

20

0.026316

21

0.026316

22

0.026316

23

0.026316

24

0.026316

25

0.026316

26

0.026316

27

0.026316

28

0.026316

29

0.026316

30

0.026316

31

0.026316

32

0.026316

33

0.026316

34

0.026316

35

0.026316

36

0.026316

Let’s say we have tracked 1,000 spins on a particular roulette table. We are thus expecting that each number would have appeared 1,000 x 0.026316 = 26.316 times. Do note that you would have to give or take an allowance of -1σ to 1σ based on the central limit theorem.

Here are our results based on 1,000 spins:

Number

Probability

Expected

Occurrence

00

0.0263

26.3158

13

0

0.0263

26.3158

32

1

0.0263

26.3158

32

2

0.0263

26.3158

27

3

0.0263

26.3158

39

4

0.0263

26.3158

20

5

0.0263

26.3158

33

6

0.0263

26.3158

23

7

0.0263

26.3158

10

8

0.0263

26.3158

36

9

0.0263

26.3158

29

10

0.0263

26.3158

17

11

0.0263

26.3158

38

12

0.0263

26.3158

14

13

0.0263

26.3158

11

14

0.0263

26.3158

25

15

0.0263

26.3158

20

16

0.0263

26.3158

16

17

0.0263

26.3158

11

18

0.0263

26.3158

12

19

0.0263

26.3158

28

20

0.0263

26.3158

17

21

0.0263

26.3158

45

22

0.0263

26.3158

24

23

0.0263

26.3158

43

24

0.0263

26.3158

25

25

0.0263

26.3158

10

26

0.0263

26.3158

21

27

0.0263

26.3158

43

28

0.0263

26.3158

23

29

0.0263

26.3158

42

30

0.0263

26.3158

30

31

0.0263

26.3158

18

32

0.0263

26.3158

45

33

0.0263

26.3158

40

34

0.0263

26.3158

41

35

0.0263

26.3158

17

36

0.0263

26.3158

30

To find the Chi2 value, the following formula applies:

Sum of all (observed values – expected values)2 / expected values

This works out to be the following:

Number

Probability

Expected

Occurrence

Observed – Expected

Observed – Expected2

Observed – Expected2/Expected

Sum

1

0.03

26.32

13

-13.32

177.31

6.74

172.984

1

0.03

26.32

32

5.68

32.31

1.23

1

0.03

26.32

32

5.68

32.31

1.23

2

0.03

26.32

27

0.68

0.47

0.02

3

0.03

26.32

39

12.68

160.89

6.11

4

0.03

26.32

20

-6.32

39.89

1.52

5

0.03

26.32

33

6.68

44.68

1.70

6

0.03

26.32

23

-3.32

10.99

0.42

7

0.03

26.32

10

-16.32

266.20

10.12

8

0.03

26.32

36

9.68

93.78

3.56

9

0.03

26.32

29

2.68

7.20

0.27

10

0.03

26.32

17

-9.32

86.78

3.30

11

0.03

26.32

38

11.68

136.52

5.19

12

0.03

26.32

14

-12.32

151.68

5.76

13

0.03

26.32

11

-15.32

234.57

8.91

14

0.03

26.32

25

-1.32

1.73

0.07

15

0.03

26.32

20

-6.32

39.89

1.52

16

0.03

26.32

16

-10.32

106.42

4.04

17

0.03

26.32

11

-15.32

234.57

8.91

18

0.03

26.32

12

-14.32

204.94

7.79

19

0.03

26.32

28

1.68

2.84

0.11

20

0.03

26.32

17

-9.32

86.78

3.30

21

0.03

26.32

45

18.68

349.10

13.27

22

0.03

26.32

24

-2.32

5.36

0.20

23

0.03

26.32

43

16.68

278.36

10.58

24

0.03

26.32

25

-1.32

1.73

0.07

25

0.03

26.32

10

-16.32

266.20

10.12

26

0.03

26.32

21

-5.32

28.26

1.07

27

0.03

26.32

43

16.68

278.36

10.58

28

0.03

26.32

23

-3.32

10.99

0.42

29

0.03

26.32

42

15.68

245.99

9.35

30

0.03

26.32

30

3.68

13.57

0.52

31

0.03

26.32

18

-8.32

69.15

2.63

32

0.03

26.32

45

18.68

349.10

13.27

33

0.03

26.32

40

13.68

187.26

7.12

34

0.03

26.32

41

14.68

215.63

8.19

35

0.03

26.32

17

-9.32

86.78

3.30

36

0.03

26.32

30

3.68

13.57

0.52

So, our Chi2 is 172.984.

We now look for this figure on the chi2 table. The numbers at the top are the probability percentages and the column on the left marked df just refer to the number of categories you are looking at -1. In our case, it’ll be 38-1 or 37. We’ll look at df=40 as it is the closest to 37.

Our chi value of 172.984 exceeds all the probabilities on the chi2 table, which means that there is DEFINITELY a PROBABILITY of the wheel or the dealer being or doing something out of the ordinary.

So, which ones?

Residuals

We can look at each number from 00 – 36 and identify which ones were out of the ordinary. We do this by calculating what is called a residual. This is just fancy for how different each result is from the expected. Here’s the formula:

Observed result – Expected result / Square root (Expected result)

So, for 00, the formula would translate to:

13 – 26.32 / square root(26.32) = -13.32/5.129 = -2.59573

Number

Probability

Expected

Occurrence

Observed – Expected

Residual

0-0

0.03

26.32

13

-13.32

-2.59573

0

0.03

26.32

32

5.68

1.108057

1

0.03

26.32

32

5.68

1.108057

2

0.03

26.32

27

0.68

0.133377

3

0.03

26.32

39

12.68

2.472608

4

0.03

26.32

20

-6.32

-1.23117

5

0.03

26.32

33

6.68

1.302993

6

0.03

26.32

23

-3.32

-0.64637

7

0.03

26.32

10

-16.32

-3.18053

8

0.03

26.32

36

9.68

1.8878

9

0.03

26.32

29

2.68

0.523249

10

0.03

26.32

17

-9.32

-1.81598

11

0.03

26.32

38

11.68

2.277672

12

0.03

26.32

14

-12.32

-2.40079

13

0.03

26.32

11

-15.32

-2.9856

14

0.03

26.32

25

-1.32

-0.25649

15

0.03

26.32

20

-6.32

-1.23117

16

0.03

26.32

16

-10.32

-2.01092

17

0.03

26.32

11

-15.32

-2.9856

18

0.03

26.32

12

-14.32

-2.79066

19

0.03

26.32

28

1.68

0.328313

20

0.03

26.32

17

-9.32

-1.81598

21

0.03

26.32

45

18.68

3.642223

22

0.03

26.32

24

-2.32

-0.45143

23

0.03

26.32

43

16.68

3.252351

24

0.03

26.32

25

-1.32

-0.25649

25

0.03

26.32

10

-16.32

-3.18053

26

0.03

26.32

21

-5.32

-1.03624

27

0.03

26.32

43

16.68

3.252351

28

0.03

26.32

23

-3.32

-0.64637

29

0.03

26.32

42

15.68

3.057415

30

0.03

26.32

30

3.68

0.718185

31

0.03

26.32

18

-8.32

-1.62105

32

0.03

26.32

45

18.68

3.642223

33

0.03

26.32

40

13.68

2.667544

34

0.03

26.32

41

14.68

2.86248

35

0.03

26.32

17

-9.32

-1.81598

36

0.03

26.32

30

3.68

0.718185

All the residuals are now in terms of σ! If you recall, the central limit theorem infers that all data should fall within the -3σ to 3σ region in relation to the mean.